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34
Computational and numerical methods for bioelectric field problems
 Critical Reviews in BioMedical Engineering
, 1997
"... Fundamental problems in electrophysiology can be studied by computationally modeling and simulating the associated microscopic and macroscopic bioelectric fields. To study such fields computationally, researchers have developed a number of numerical and computational techniques. Advances in computer ..."
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Fundamental problems in electrophysiology can be studied by computationally modeling and simulating the associated microscopic and macroscopic bioelectric fields. To study such fields computationally, researchers have developed a number of numerical and computational techniques. Advances in computer architectures have allowed researchers to model increasingly complex biophysical system. Modeling such systems requires a researcher to apply a wide variety of computational and numerical methods to describe the underlying physics and physiology of the associated threedimensional geometries. Issues naturally arise as to the accuracy and efficiency of such methods. In this paper we review computational and numerical methods for solving bioelectric field problems. The motivating applications represent a class of bioelectric field problems that arise in electrocardiography and
A nondiffusive finite volume scheme for the threedimensional Maxwell's equations on unstructured meshes
 SIAM J. Num. Anal
, 2002
"... Abstract. We prove a sufficient CFLlike condition for the L2stability of the secondorder accurate finite volume scheme proposed by Remaki for the timedomain solution of Maxwell’s equations in heterogeneous media with metallic and absorbing boundary conditions. We yield a very general sufficient ..."
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Abstract. We prove a sufficient CFLlike condition for the L2stability of the secondorder accurate finite volume scheme proposed by Remaki for the timedomain solution of Maxwell’s equations in heterogeneous media with metallic and absorbing boundary conditions. We yield a very general sufficient condition valid for any finite volume partition in two and three space dimensions. Numerical tests show the potential of this original finite volume scheme in one, two, and three space dimensions for the numerical solution of Maxwell’s equations in the timedomain.
The Scirun Problem Solving Environment And Computational Steering Software System
, 1999
"... Since the introduction of computers, scientists and engineers have attempted to harness their power to simulate complex physical phenomena. Today, the computer is an almost universal tool used in a wide range of scientific and engineering domains. Currently, organizing, running and visualizing a ne ..."
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Cited by 18 (6 self)
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Since the introduction of computers, scientists and engineers have attempted to harness their power to simulate complex physical phenomena. Today, the computer is an almost universal tool used in a wide range of scientific and engineering domains. Currently, organizing, running and visualizing a new largescale simulation still requires hours or days of a researcher's time. Time and effort required for data input, output and conversion further slows and complicates process. We present the design and application of SCIRun, a Problem Solving Environment (PSE), and a computational steering software system. SCIRun allows a scientist or engineer to interactively steer a computation, changing parameters, recomputing, and then revisualizing all within the same programming environment. The tightly integrated modular environment provided by SCIRun allows computational steering to be applied to a broad range of advanced scientific computations. This dissertation demonstrates that computationa...
Bifurcations of hyperbolic planforms
 Journal of Nonlinear Science
, 2011
"... Motivated by a model for the perception of textures by the visual cortex in primates, we analyse the bifurcation of periodic patterns for nonlinear equations describing the state of a system defined on the space of structure tensors, when these equations are further invariant with respect to the iso ..."
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Motivated by a model for the perception of textures by the visual cortex in primates, we analyse the bifurcation of periodic patterns for nonlinear equations describing the state of a system defined on the space of structure tensors, when these equations are further invariant with respect to the isometries of this space. We show that the problem reduces to a bifurcation problem in the hyperbolic plane D (Poincaré disc). We make use of the concept of periodic lattice in D to further reduce the problem to one on a compact Riemann surface D/Γ, where Γ is a cocompact, torsionfree Fuchsian group. The knowledge of the symmetry group of this surface allows to carry out the machinery of equivariant bifurcation theory. Solutions which generically bifurcate are called ”Hplanforms”, by analogy with the ”planforms ” introduced for pattern formation in Euclidean space. This concept is applied to the case of an octagonal periodic pattern, where we are able to classify all possible Hplanforms satisfying the hypotheses of the Equivariant Branching Lemma. These patterns are however not straightforward to compute, even numerically, and in the last section we describe a method for computation illustrated with a selection of images of octagonal Hplanforms.
Adaptive eigenvalue computation: Complexity estimates
 Numer. Math
"... This paper is concerned with the design and analysis of a fully adaptive eigenvalue solver for linear symmetric operators. After transforming the original problem into an equivalent one formulated on ℓ2, the space of square summable sequences, the problem becomes sufficiently well conditioned so tha ..."
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Cited by 10 (4 self)
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This paper is concerned with the design and analysis of a fully adaptive eigenvalue solver for linear symmetric operators. After transforming the original problem into an equivalent one formulated on ℓ2, the space of square summable sequences, the problem becomes sufficiently well conditioned so that a gradient type iteration can be shown to reduce the error by some fixed factor per step. It then remains to realize these (ideal) iterations within suitable dynamically updated error tolerances. It is shown under which circumstances the adaptive scheme exhibits in some sense
Mathematical modeling of secondary lithium batteries
 J. Electrochem. Acta
, 2000
"... Modeling of secondary lithium batteries is reviewed in this paper. The models available to simulate the electrochemical and thermal behavior of secondary lithium batteries are discussed considering not only their electrochemical representation (transport phenomena and thermodynamics of the system), ..."
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Cited by 9 (0 self)
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Modeling of secondary lithium batteries is reviewed in this paper. The models available to simulate the electrochemical and thermal behavior of secondary lithium batteries are discussed considering not only their electrochemical representation (transport phenomena and thermodynamics of the system), but also the mathematical techniques that have been used for solving the equations. A brief review of the governing equations for porous electrodes, and methods for solving these equations is also given. © 2000 Elsevier Science Ltd. All rights reserved.
LeastSquares FiniteElement Solution Of The Neutron Transport Equation In Diffusive Regimes
, 1998
"... A systematic solution approach for the neutron transport equation, based on a leastsquares finiteelement discretization, is presented. This approach includes the theory for the existence and uniqueness of the analytical as well as of the discrete solution, bounds for the discretization error, and g ..."
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Cited by 8 (1 self)
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A systematic solution approach for the neutron transport equation, based on a leastsquares finiteelement discretization, is presented. This approach includes the theory for the existence and uniqueness of the analytical as well as of the discrete solution, bounds for the discretization error, and guidance for the development of an efficient multigrid solver for the resulting discrete problem. To guarantee the accuracy of the discrete solution for diffusive regimes, a scaling transformation is applied to the transport operator prior to the discretization. The key result is the proof of the V  ellipticity and continuity of the scaled leastsquares bilinear form with constants that are independent of the total cross section and the absorption cross section. For a variety of leastsquares finiteelement discretizations this leads to error bounds that remain valid in diffusive regimes. Moreover, for problems in slab geometry a full multigrid solver is presented with V (1, 1)cycle convergence factors approximately equal to 0.1 independent of the size of the total cross section and the absorption cross section.
On the Numerical Evaluation of Fractional Sobolev Norms
"... Abstract. In several important and active fields of modern applied mathematics, such as the numerical solution of PDEconstrained control problems or various applications in image processing and data fitting, the evaluation of (integer and real) Sobolev norms constitutes a crucial ingredient. Diffe ..."
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Abstract. In several important and active fields of modern applied mathematics, such as the numerical solution of PDEconstrained control problems or various applications in image processing and data fitting, the evaluation of (integer and real) Sobolev norms constitutes a crucial ingredient. Different approaches exist for varying ranges of smoothness indices and with varying properties concerning exactness, equivalence and the computing time for the numerical evaluation. These can usually be expressed in terms of discrete Riesz operators. We propose a collection of criteria which allow to compare different constructions. Then we develop a unified approach which is valid for nonnegative real smoothness indices for standard finite elements, and for positive and negative real smoothness for biorthogonal wavelet bases. This construction delivers a wider range of exactness than the currently known constructions and is computable in linear time. 1. Introduction. Sobolev
Finite element approximations for Schrödinger equations with applications to electronic structure computations
 J. Comput. Math
"... Dedicated to the 70th birthday of Professor Junzhi Cui In this paper, both the standard finite element discretization and a twoscale finite element discretization for Schrödinger equations are studied. The numerical analysis is based on the regularity that is also obtained in this paper for the Sc ..."
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Cited by 7 (5 self)
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Dedicated to the 70th birthday of Professor Junzhi Cui In this paper, both the standard finite element discretization and a twoscale finite element discretization for Schrödinger equations are studied. The numerical analysis is based on the regularity that is also obtained in this paper for the Schrödinger equations. Very satisfying applications to electronic structure computations are provided, too.